# What is $\tilde{\Bbb{C}}$

$\tilde{\Bbb{C}}$ was defined in the following manner

$\tilde{\Bbb{C}} = \Bbb{R} \cdot 1 + \Bbb{R} \cdot e$

with $1 \cdot 1 = 1, 1 \cdot e = e \cdot 1, e \cdot e = 1$

Could you elaborate more on this please? Or simply what is its name?

There are three ways of extending multiplication to the plane which are useful in elementary geometry. One writes a planar vector formally as $a+be$ for some new element $e$. The arithmetic is then determined by knowledge of $e^2$. The best known is the case of the complex numbers ($e^2=-1$) (related to eulidean geometry). The case $e^2=0$ leads to the so-called dual numbers. Finally, $e^2=1$ is appropriate for hyperbolic geometry. A suitable reference is Yaglom "Complex numbers in geometry". Interesting exercise: compute the exponential function $\exp(a+b e)$ using the power series expansion of the exponential function in the other two cases.
A way to think about this set is $\mathbb{R}[\sqrt{1}]$ (in contrast to $\mathbb{C}=\mathbb{R}[\sqrt{-1}]$. $e$ here is a formal symbol whose square is 1, just like $i$ is a formal symbol whose square is $-1$. A little more formally is that this is isomorphic to $\mathbb{R}[x]/(x^2-1)$, where $(x^2-1)$ is the ideal generated by $x^2-1$.
• Yeah, just realized that. Edited, along with a comparison of this field and $\mathbb{C}$ – Stella Biderman Apr 3 '14 at 15:23
• What is the difference between $\mathbb{R}[\sqrt{1}]$ and $\Bbb{R}$ – george Apr 3 '14 at 16:21
• That notation is used to draw attention to the fact that $e$ is a formal symbol which acts like a square root of 1 that doesn't lie within $\mathbb{R}$. This is a notation that I don't think is formally correct but I have encountered in number theory – Stella Biderman Apr 3 '14 at 18:29